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Theorem exists1 2359
Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4612. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2269 . 2  |-  ( E! x  x  =  x  <->  E. y A. x ( x  =  x  <->  x  =  y ) )
2 equid 1840 . . . . . 6  |-  x  =  x
32tbt 345 . . . . 5  |-  ( x  =  y  <->  ( x  =  y  <->  x  =  x
) )
4 bicom 203 . . . . 5  |-  ( ( x  =  y  <->  x  =  x )  <->  ( x  =  x  <->  x  =  y
) )
53, 4bitri 252 . . . 4  |-  ( x  =  y  <->  ( x  =  x  <->  x  =  y
) )
65albii 1687 . . 3  |-  ( A. x  x  =  y  <->  A. x ( x  =  x  <->  x  =  y
) )
76exbii 1712 . 2  |-  ( E. y A. x  x  =  y  <->  E. y A. x ( x  =  x  <->  x  =  y
) )
8 nfae 2111 . . 3  |-  F/ y A. x  x  =  y
9819.9 1943 . 2  |-  ( E. y A. x  x  =  y  <->  A. x  x  =  y )
101, 7, 93bitr2i 276 1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435   E.wex 1659   E!weu 2265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-eu 2269
This theorem is referenced by:  exists2  2360
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